Fixed points for multi-class queues

Burke's theorem can be seen as a fixed-point result for an exponential single-server queue; when the arrival process is Poisson, the departure process has the same distribution as the arrival process. We consider extensions of this result to multi-type queues, in which different types of customer have different levels of priority. We work with a model of a queueing server which includes discrete-time and continuous-time M/M/1 queues as well as queues with exponential or geometric service batches occurring in discrete time or at points of a Poisson process. The fixed-point results are proved using interchangeability properties for queues in tandem, which have previously been established for one-type M/M/1 systems. Some of the fixed-point results have previously been derived as a consequence of the construction of stationary distributions for multi-type interacting particle systems, and we explain the links between the two frameworks. The fixed points have interesting "clustering" properties for lower-priority customers. An extreme case is an example of a Brownian queue, in which lower-priority work only occurs at a set of times of measure 0 (and corresponds to a local time process for the queue-length process of higher priority work).Comment: 25 page

Some applications of geometric process model.

by Kit-ching To.Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.Includes bibliographical references (leaves 64-67).Abstract also in Chinese.Chapter Chapter 1 --- Overview --- p.1Chapter Chapter 2 --- Geometric Process Model --- p.6Chapter 2.1 --- Introduction --- p.6Chapter 2.2 --- Some Properties of Geometric Process --- p.7Chapter 2.3 --- Geometric Process Model for the Exponential Distribution --- p.13Chapter Chapter 3 --- Analysis of Warranty Policy by a Geometric Process Model --- p.16Chapter 3.1 --- Introduction --- p.16Chapter 3.2 --- Consumer's Policy --- p.17Chapter 3.2.1 --- N-Repair Warranty Policy --- p.19Chapter 3.2.2 --- n-Renewal Warranty Policy --- p.23Chapter 3.2.2.1 --- Modified n-Renewal Warranty Policy --- p.25Chapter 3.2.3 --- Optimal Policy to Consumer --- p.29Chapter 3.3 --- Manufacturer's Policy --- p.33Chapter 3.3.1 --- Optimal Policy to Manufacturer --- p.36Chapter 3.4 --- Numerical Examples --- p.38Chapter Chapter 4 --- Analysis of a Repairable M/M/l Queueing System by a Geometric Process Model --- p.45Chapter 4.1 --- Introduction --- p.45Chapter 4.2 --- Model --- p.46Chapter 4.2.1 --- Some Reliability Indices --- p.53Chapter 4.2.1.1 --- Availability --- p.53Chapter 4.2.1.2 --- Rate of Occurrence of Failures (ROCOF) --- p.54Chapter 4.3 --- Numerical Method --- p.56Chapter 4.3.1 --- Results --- p.58Bibliography --- p.6

An analytical model for gas overpressure in slug-driven explosions:insights into Strombolian volcanic eruptions

Strombolian eruptions, common at basaltic volcanoes, are mildly explosive events that are driven by a large bubble of magmatic gas (a slug) rising up the conduit and bursting at the surface. Gas overpressure within the bursting slug governs explosion dynamics and vigor and is the main factor controlling associated acoustic and seismic signals. We present a theoretical investigation of slug overpressure based on magma-static and geometric considerations and develop a set of equations that can be used to calculate the overpressure in a slug when it bursts, slug length at burst, and the depth at which the burst process begins. We find that burst overpressure is controlled by two dimensionless parameters: V', which represents the amount of gas in the slug, and A', which represents the thickness of the film of magma that falls around the rising slug. Burst overpressure increases nonlinearly as V' and A' increase. We consider two eruptive scenarios: (1) the "standard model," in which magma remains confined to the vent during slug expansion, and (2) the " overflow model," in which slug expansion is associated with lava effusion, as occasionally observed in the field. We find that slug overpressure is higher for the overflow model by a factor of 1.2-2.4. Applying our model to typical Strombolian eruptions at Stromboli, we find that the transition from passive degassing to explosive bursting occurs for slugs with volume >24-230 m(3), depending on magma viscosity and conduit diameter, and that at burst, a typical Strombolian slug (with a volume of 100-1000 m(3)) has an internal gas pressure of 1-5 bars and a length of 13-120 m. We compare model predictions with field data from Stromboli for low-energy " puffers," mildly explosive Strombolian eruptions, and the violently explosive 5 April 2003 paroxysm. We find that model predictions are consistent with field observations across this broad spectrum of eruptive styles, suggesting a common slug-driven mechanism; we propose that paroxysms are driven by unusually large slugs (large V')

A New Methodology for Recognition of Milling Features from STEP File

In recent years, various researchers have come up with different ways and means to integrate CAD and CAM. Automatic feature recognition (AFR) from a CAD solid model for down stream applications like process planning and NC program, greatly contribute to the level of integration. When generating G&M codes from CAD DXF file, it leads to the loss of geometric information and the user is to edit and fills the details of the lost data. STEP is an international standard for geometric and non geometric data transfer between CAD, CAE and CAM and it replaces the IGES and DXF. For that reason this paper proposes an automatic feature recognition methodology to develop a feature recognition system using STEP file. The proposed methodology is developed for 3D prismatic parts that are modeled any CAD software having STEP output file format. A JAVA program is used to implement the geometric data extraction algorithm, which has been developed for extracting the geometric information from the STEP file. A feature recognition algorithm is used to recognize the different features of the part such as slot, pocket etc based on geometric reasoning approach by taking B-rep data base as input. The authors present an example to demonstrate the application of the proposed methodology

Multiple scattering in random mechanical systems and diffusion approximation

This paper is concerned with stochastic processes that model multiple (or iterated) scattering in classical mechanical systems of billiard type, defined below. From a given (deterministic) system of billiard type, a random process with transition probabilities operator P is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions. Of particular interest are systems with weak scattering, which are associated to parametric families of operators P_h, depending on a geometric or mechanical parameter h, that approaches the identity as h goes to 0. It is shown that (P_h -I)/h converges for small h to a second order elliptic differential operator L on compactly supported functions and that the Markov chain process associated to P_h converges to a diffusion with infinitesimal generator L. Both P_h and L are selfadjoint (densely) defined on the space L2(H,{\eta}) of square-integrable functions over the (lower) half-space H in R^m, where {\eta} is a stationary measure. This measure's density is either (post-collision) Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes with infinitesimal generator L respectively correspond to what we call MB diffusion and (generalized) Legendre diffusion. Concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes.Comment: 34 pages, 13 figure